Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2012, Article ID 871091, 10 pages
doi:10.1155/2012/871091
Research Article
An Optimization of Tree Topology Based
Parallel Cryptography
Masumeh Damrudi and Norafida Ithnin
Department of Computer System and Communication, Faculty of Computer Science and
Information Systems, Universiti Teknologi Malaysia, Skudai, 81310 Johor Bahru, Malaysia
Correspondence should be addressed to Masumeh Damrudi, m.damrudi@gmail.com
Received 26 April 2012; Accepted 2 August 2012
Academic Editor: Wanquan Liu
Copyright q 2012 M. Damrudi and N. Ithnin. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Public key cryptography has become of vital importance regarding the rapid development of
wireless technologies. The RSA is one of the most important algorithms for secure communications
in public-key cryptosystems. Since the RSA is expensive in terms of computational task which is
modular exponentiation, parallel processing and architecture is a reasonable solution to speedup
RSA operations. In this paper, taking into account pipelining and optimization, we improve
throughput and efficiency of the TRSA method, a parallel architecture solution for RSA security
based on tree topology. The optimization and pipelining of the tree based architecture increases
its efficiency and throughput. The experimental results demonstrate that these pipelined and
optimized approaches outperform the main TRSA.
1. Introduction
Using wireless communication has made most of today’s systems vulnerable. Employing
appropriate measures, which provide confidentiality, integrity, authenticity, and availability
for all messages in presence of adversaries, can reduce threats on applications exploiting
wireless communications such as WSN. Asymmetric algorithms called public key can
provide user authentication and be useful in key distribution and management. Public key
encryption allows two parties to communicate secretly, even if all communications between
them are monitored. Furthermore, it allows enormous flexibility, which in example is essential for an online merchant where an online merchant processing credit card orders from multiple purchasers. It is more convenient for the receiver to store a single private key rather than
to share and manage different secret keys.
Once asymmetric cryptographies are based on number theory, they are expensive in
terms of their mathematical calculations 1, 2, which need large amount of energy resources
2
Mathematical Problems in Engineering
and sufficient amount of memory for large keys 3. Benefiting from software solutions for
cryptography leads to flexibility and ease of use and upgrade whilst it do not provide enough
speed and are less secure than hardware solutions. Software issues can be easily monitored
and on the other hand they consume a lot of resources. Besides, it is difficult to transfer them
among different operating systems whilst hardware methods need fewer computer resources
and have special chips to accelerate the process 4. As a real-world example, the security
problem in sensor networks is more challenging due to the resource limited nodes. In such
low resource devices, a solution based on parallel architecture becomes more appropriate due
to the smaller occupied area by the architecture 5. Basically, energy efficiency and battery
life time play a major role in the lifetime of such applications 6.
The main reason of using topological interconnection of processor elements in terms
of parallel architecture is to create a powerful computer or processing element for specific
objectives. Parallel architecture is combined of nodes and these interconnection networks.
Depending on the algorithm being deployed on the architecture, either of pipelining or optimization and in some cases both of them are applicable to the solutions. While the pipelined
approach increases the throughput by processing multiple PEs simultaneously, the optimization improves efficiency by eliminating the redundant PEs.
The main contribution of this paper lies on using parallel pipelined of Tree topology for
RSA cryptography and issuing an optimization to this work. The pipelining, and moreover,
optimization improve the results drastically.
The rest of this paper is organized as follows. Firstly, we summarize the related works
in the next section. Then, the pipelined TRSA is explained. Section 4 provides the optimization of TRSA. Afterwards, the simulation results are discussed in Section 5. Finally, the conclusion of the work is drawn.
2. Related Works
We have summarized existing parallel approaches on cryptographic algorithms 7. In addition, a parallel cryptography method using RSA is proposed in 8, which uses tree topology
as its base infrastructure. As far as the authors’ knowledge, from the topology point of view,
there is no significant parallel approach on RSA algorithm other than TRSA. The well-known
parallel methods using software and hardware solutions on RSA are in 5, 9–21. More details
on these approaches are presented in 7. Among these parallel approaches, the best value in
terms of seconds is 3.23 ms while applying 1024 bit key length 5. The greatest key length is
3072 12, and the best speedup is 10.9 using 2560 key length 13. The evaluation metric in
these methods is speedup or time for most of the cases, and hardware implementations have
employed Montgomery’s algorithm to perform modular multiplication or exponentiation.
These recent approaches have not discussed the time complexity or the order of the
algorithms which are inseparable from parallel processing. To the best of the authors’ knowledge, the only existing discussion on time complexity is the known CRT 22, Montgomery
23, and the binary 22 which is an accepted method, that are based on the number of
multiplications. Herewith, to analyze the proposed approach, we deal with the time complexity of optimization and pipelining of TRSA method.
3. Pipelined TRSA
TRSA is a new parallel approach on RSA using the tree interconnection network 8. TRSA
can be applied as a coprocessor for embedded systems such as wireless sensor nodes, which
Mathematical Problems in Engineering
3
require more speed in transferring confidential information. Enhancing TRSA, pipelining
mechanism is employed to achieve higher throughput. Applying pipelining to the main solution, the latency between data that are being encrypted drastically decreases 5. It is assumed
that the base operation is one multiplication and one modulus, which is called MulMod in
brief. Considering the encryption as Ci mei mod n where i is the block number, in TRSA,
computing Ci should be finished before the Ci1 starts to be computed. In this enhancement to
the main approach, when the results of current operations are sent to the next level of tree, the
computation for next ciphertext can be started in this level. In principle, a new operation can
be initiated with this frequency. Even though previous ones are still in pipeline, and the
ciphertext is not ready for prior plaintext so far.
The following definitions are used in the original algorithm.
np : The number of PEs of tree 1
el : The exponentiation for the last leaf node’s input
eo : The exponentiation for other leaf nodes’ input
pi : ith processor element.
The RSA consists of two great prime numbers p and q, which compute modulus n pq. The value ϕ p−1q−1 is employed to determine e, where e < n and gcde, ϕ 0. Consequently, d has the following relation:
d e − 1 mod ϕ.
3.1
Let eo , el , A, and B be computed as following:
eo e div np ,
3.2
el eo e mod np ,
3.3
A meo mod n,
3.4
B mel mod n.
3.5
Figure 1a presents the schematic structure of a tree-based architecture using seven PEs.
Figure 1b demonstrates inner structure of each PE.
The encryption/decryption solution in TRSA using seven PEs as tree topology is
indicated in the following. Processor elements from p0 to p2 receive the value of A as their
inputs, and p3 will receive both A and B. These PEs will perform the following operations:
∀i, ∈ {i ≥ 0, i ≤ 2};
pi computes A × A mod n,
p3 computes A × B mod n.
3.6
4
Mathematical Problems in Engineering
P6
Level 1
Level 2
Level 3
P4
P0
P5
P1
A
A
A
Coordinator
P3
P2
A
B
P
n
in1
in2
(a)
mod
out
×
(b)
Figure 1: a Processor elements in tree architecture, b interior structure of each PE.
PEs p0 to p3 will send their results to their parents which are p4 and p5 . The values D
and E are the inputs of p4 and p5 that are computed as follows:
D A × A mod n,
E A × B mod n.
3.7
In the second step, processor elements p4 and p5 will carry out the following operations
to compute F and G:
p4 computes F D × D mod n,
p5 computes G D × E mod n.
3.8
Then p4 and p5 will send their results to the root. The root will execute the following
operation to generate the output of p6 which is the encryption of m using RSA:
p6 computes F × G mod n
3.9
The details of the operations are discussed in 8.
The pipeline phases are shown in Figure 2. It is shown that while the computation for
C1 is still in process, the computations for other messages have already been started.
The number of levels depends on the number of nodes in the tree. In this example,
number of nodes is eight in which seven of them are forming the tree, and the other one is
the coordinator. However, the number of nodes can be more or less but not less than four.
Number of PEs should follow the tree rule which is 2No. of levels − 1, in addition a processor
element is also added as the coordinator of first operation. Using more nodes, the more parallelization will be gained, nevertheless, overloading of more processor elements for smaller
data and key length should be prevented. The coordinator operations will be decreased due to
the increase in the number of PEs for the tree which leads to less execution time. In this
approach, there is always a tradeoff between the number of PEs, the data size, and the key
length.
Mathematical Problems in Engineering
P6
P6
P4
A
C1
P5
P1
P0
P2
A A A
P
5
P4
P3
C1
B
A
C1
P5
P1
P0
P2
A A A
P3
a
C2
P4
A
P2
A A A
C3
b
P
c
P6
C2
P5
P1
P0
B
P
C2
C1
P6
P3
C3
P4
P1
P0
A
B
C4
P5
P2
A A A
P3
B
P
d
Figure 2: Steps of pipelining in TRSA.
3.1. Analysis of Pipelining
Once levels one, two, and three do just one MulMod operation, their execution times indicated by T1 , T2 , and T3 are equal and we have
T1 T2 T3 .
3.10
The execution time of the operation in the coordinator depends on the computational
power of the coordinator and the el . Although tree architecture used here is a homogenous
architecture itself, this architecture is heterogeneous as a whole, which means that the coordinator processor element is different from other process elements. The PE known as coordinator must be more advanced than others. Assuming this and knowing that the computation in this PE is more complicated than others, the difference between the execution time
of this processor element and others should be very small. As an applied example, the coordinator can be the CPU of the embedded system, and the tree part can be used as a coprocessor. In this case, the coprocessor is homogeneous. According to the above discussion, and
using Tc as the execution time of coordinator, we can have
T1 T2 T3 ∼
Tc .
3.11
Using pipelining, as it is clarified in Figure 2, the throughput of TRSA with pipelining
is about four-times of the throughput of original TRSA for a tree with seven nodes. In the
pipelined version, when computation of C1 is completed, calculation of C5 will be started as a
new block; whilst in the original TRSA, when C1 is computed, calculation of C2 will be started
as the next block. Th is representing throughput in the subsequent equations:
T1 T2 T3 ∼
Tc ,
Tpipelined 1
Toriginal ,
4
3.12
Thpipelined 4Thoriginal .
It should be considered that if the number of processor elements increase, the
throughput of pipelining will be increased too. The relation of the throughput for
6
Mathematical Problems in Engineering
P14
P14
P 12
P8
P 12
P13
P9
P10
P8
P11
P 12
P13
P9
P10
P11
P0 P1 P2 P3 P4 P5 P6 P7
P 0 P1 P2 P 3 P4 P 5 P6 P7
P
P14
Eliminated PEs
(a)
P13
P8
P11
P0
P7
P
P
(b)
(c)
Figure 3: Optimization of TRSA using tree with 15 PEs.
the pipelined TRSA and the throughput for original TRSA where l represents the levels of
the tree is
Thpipelined l 1Thoriginal .
3.13
In this approach, all PEs are busy all the time computing cipher texts while in the
original TRSA, just one level of the processor elements was busy computing, and the others
were idle. The pipelining mechanism provides an appropriate load balancing. In this example, having just four pipeline stages has led to a higher throughput, and more pipelining by
employing a bigger tree architecture will achieve higher performance and more efficient
encryption implementation.
4. Optimization of TRSA
The original TRSA method can be improved and optimized to increase efficiency as well as
decrease area, which is one of the most important factors in energy and size limited systems
like sensor nodes. In the original TRSA there are some nodes, which perform the same operation during operation and generate the same result. Knowing this, the results of some PEs
can be employed instead of the results of some other PEs. These redundant PEs can be
eliminated. Figure 3 illustrates the transition from original TRSA to the optimized TRSA.
Considering four levels in a tree, the PEs which are doing the same operation are marked with
the same shape in Figure 3a. The nodes that can be eliminated from the tree to optimize the
algorithm are omitted in Figure 3c and new connections are created.
Eliminating redundant PEs leads to a smaller area usage. By the increase in levels of
the tree, the redundant PEs will be increased. If the levels of parallelizing are more than four,
the number of eliminated PEs increases exponentially.
4.1. Analysis of Optimization
Efficiency of the parallel approaches 24–26 is measured using efficiency factor, which is
E S/P , where P and S are the number of processors and speedup, respectively. Assuming
the number of PEs to be 15 as demonstrated in Figure 3, the efficiency of the algorithm
will increase from S/15 to S/7 employing optimization. Taking into the general form,
Mathematical Problems in Engineering
7
the efficiency of original TRSA is E S/2l − 1 where l is the number of tree levels.
Optimizing the TRSA, the efficiency of optimized TRSA becomes
Eop S
.
2l − 1
4.1
The improvement of efficiency is obtained by
l
Eop
2 −1
.
E
2l − 1
4.2
The above equation shows that the efficiency is boosted exponentially, especially, when
the number of levels increases.
The number of multiplications for accepted method, which is binary in the best case, is
k − 1 where k is the number of bits of the exponent 14, 22, 27. The number of multiplications
in the binary method for the worst and average cases are 2k − 1 and 1.5k − 1. Let l be the
number of levels, the power of A and B in 3.4 and 3.5 will be divided to the number of
PEs which is 2l . Hence, 2k − 1 − log2l multiplications will be done to compute A and B
consequently. Thus the number of multiplications of coordinator for the worst case is
2k − l − 1.
K 2 k − 1 − log 2l
4.3
The total number of multiplications for optimized TRSA is 2l − 1, thus the total
multiplication count is 2k − l − 1 2l − 1. While there are some operations which execute
simultaneously in the parallel architectures, the actual number of multiplications is much less
than the total number of multiplications. In this case, the total number of PEs in optimized
TRSA is 2l − 1 where l is the level of tree and the number of multiplications is l due to the
concurrency of PEs. Therefore, the total number of multiplications for encryption is decreased
to 2k − l − 1 l.
Using the calculation method from 11, the number of multiplications is presented
with ηk, l and in result the number of multiplications for the optimized TRSA in the worst
case is
ηk, l 2k − 1 − l.
4.4
Table 1 compares the number of multiplications in the best, average, and worst case
for binary, CRT, Montgomery, and optimized TRSA for both general form and a tree with 60
levels.
The number of multiplications of primitive RSA will be reduced using the optimized
TRSA method compared to the CRT and Montgomery method. Although, the best case in the
TRSA and the binary methods are the same, the average and worst cases of TRSA are
improved. The level of this improvement depends on the number of MulMod blocks in the
tree topology and the length for RSA cryptographic key in bits.
8
Mathematical Problems in Engineering
Table 1: Number of multiplications in binary, CRT, montgomery, and optimized TRSA.
Method
Best case
Average case
Worst case
General form k 1024 l 60 General form k 1024 l 10 General form k 1024 l 60
Binary
k−1
1023
1.5k − 1
1535
2k − 1
2046
TRSA
k−1
1023
1.5k − 1 − 0.5l
1505
2k − 1 − l
1986
787456
CRT
3k 2 /4 k
2098176
Montgomery
2k 2 k
18
11
16
10
14
Efficiency (%)
Throughput
9
8
7
6
10
8
6
4
5
4
12
2
3
4
5
6
7
8
9
10
0
1000
Number of tree levels
1500
2000
2500
3000
3500
4000
4500
Key length (bit)
TRSA
Optimized TRSA
a
b
Figure 4: Optimization and pipelining of TRSA. a Throughput of original TRSA versus pipelined TRSA.
b Efficiency of original TRSA versus Optimized TRSA.
5. Simulations, Results, and Discussion
Benefiting from pipelining technique and the optimization method, the variation of TRSA
presented in this paper outperforms original TRSA. According to 8 and the results of time
complexity that is based on the number of multiplications, original TRSA also outperforms
the well-known existing approaches from the literature which are CRT, Montgomery, Binary,
and the sequential approaches.
The advantages of pipelining in terms of throughput are presented in Figure 4a. The
throughput is Thpipelined l 1Thoriginal TRSA . The advantages of optimization in terms of
efficiency are presented in Figure 4b.
Considering the number of levels for TRSA to be three, the throughput of pipelined
TRSA is four-times of main TRSA. The more levels employed, the better throughput obtained.
The simulation results of optimization for the average of 100 iterations are utilized to
draw Figure 4b. The results are achieved using C language of Microsoft Visual Studio
2008, OpenMP and OpenSSL. The efficiency is E S/P where S is representing the execution
time for TRSA and optimized TRSA, and the efficiency is multiplied by 100 to be in percent
unit. When the number of PEs for original TRSA is seven, the number of PEs in Optimized
TRSA becomes five. Just like the case in pipelining, the more levels of tree employed, the
more efficiency obtained.
Mathematical Problems in Engineering
9
Figure 4b shows the efficiency of the original TRSA versus optimized TRSA for 1024,
2048, and 4096 bit key lengths using 64, 128, and 256 byte input file and 61, 29, and 15 levels,
respectively. The optimized TRSA has preferred efficiency than original TRSA, which means
that less area is required.
Selecting number of levels for a tree among others depends on the area and the speed
which is desired for the security needs of the target system. The speed and number of PEs are
always related to each other.
The optimized TRSA has more efficiency than original TRSA, which means that it
requires less area than TRSA that leads to less hardware complexity. It is necessary to clarify
that the two variations of the original TRSA represented in this paper are not in conflict with
each other. Taking pipelining into account along with the optimization, the outcome will be
optimum in terms of speed, throughput, and efficiency all at the same time as three significant
parameters.
6. Conclusions
This paper has come up with a pipelining and optimization approach for original TRSA to
achieve better results in RSA encryption using TRSA method. The efficiency of the original
TRSA approach is improved using the optimization whilst the throughput is increased by
conducting pipelining. The increase in the efficiency resulted from optimization of TRSA
depends on the bits of key and the tree levels. Just like efficiency, the throughput also
increases by the increase in the tree levels. The simulation results confirmed that having a
greater key indicates the efficiency of optimized TRSA which becomes better than original
TRSA. Applying the optimization, the number of PEs decreases, which results in less area
usage that is extremely important in size and energy limited embedded systems.
Acknowledgment
The authors do not have any conflict of interests with the content of the paper.
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